Let's analyze the graph and address both questions (a) and (b):

Part (a): $f(1)$

From the graph, observe the point where $x = 1$. At $x = 1$, the graph has a hollow dot, indicating that the function $f(x)$ is not defined at this point. Therefore:

$f(1) = \text{undefined}$

Part (b): $\lim_{x \to 1} f(x)$

To evaluate the limit $\lim_{x \to 1} f(x)$, we need to consider the behavior of the function as $x$ approaches 1 from both the left-hand side (as $x \to 1^-$) and the right-hand side (as $x \to 1^+$).

• As $x \to 1^-$ (from the left), the graph approaches the value $2$.
• As $x \to 1^+$ (from the right), the graph also approaches the value $2$.

Since the left-hand limit and right-hand limit are equal, we conclude that the two-sided limit exists, and:

$\lim_{x \to 1} f(x) = 2$

Summary:

• $f(1)$ is undefined.
• $\lim_{x \to 1} f(x) = 2$.

Would you like any more details or explanations?

Here are five related questions to expand your understanding:

1. What is the difference between a limit and a function's value at a point?
2. How do you evaluate one-sided limits using a graph?
3. What conditions must hold for a limit to exist at a given point?
4. How can a function have a limit at a point where it is undefined?
5. What happens if the left-hand limit and right-hand limit do not match?

Tip: When analyzing a graph, always check for discontinuities, as they play a crucial role in determining whether the function is defined or the limit exists at specific points.